How did the Soviet Union respond to the U.S. policy of brinkmanship?

Through defensive policies maintaining control of Eastern Europe and involvement in the 3rd World conflicts.

It depends what is meant by brinkmanship. This is usually associated with the Cuban Missile Crisis and applies to the Soviet Union as much as the USA. In this instance, it was the Soviets who backed down and it led to Khruschev's demise.

If brinkmanship means U.S foreign policy designed to extend American control throughout the world and specifically against the Soviet Union in Europe, then the Soviet reaction was as follows. In conflicts such as Vietnam and the Middle East, the Soviets supported governments and factions which opposed US backed governments and factions. Examples include North Vietnam, Egypt and Syria.

In Africa the Soviets were heavily involved in in Angola and indeed Mozambique. They also supported Somalia in its disputes with Ethiopia and in an example of extreme irony/hypocrisy both the Soviet Union and the USA switched sides after the fall of Haile Selasse in Ethiopia.

In Europe the Warsaw Pact established a collective security alliance protecting the Soviet Union from an attack by the West.